Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

The Newton–wave prescription

Model 3: a specified sourced form and an exterior pairing › The Newton–wave prescription · lines 1098–1133

IV-B@1#eq:NW

Translate $t_{0}$ to zero. On the completion, let $Q_{μν}$ be a smooth conserved symmetric source, Schwartz in space with all seminorms locally uniform in time. Define

(Δ^{- 1} b) (x) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d y, ρ = Q_{00} (0), J_{i} = Q_{0 i} (0),

and

a = - 2 Δ^{- 1} ρ, V_{i} = 2 Δ^{- 1} J_{i} .

For trace-reversed data $A_{μν}, B_{μν}$ , prescribe

A_{00} = a, A_{0 i} = - V_{i}, A_{ij} = 0, B_{00} = - \partial_{i} V_{i}, B_{0 i} = B_{ij} = 0.

With $D_{0} = - Δ$ , put

j (t) = cos (t D_{0}) A + \frac{sin ( t D _{0} )}{D _{0}} B + 2 \int_{0}^{t} \frac{sin (( t - s ) D _{0} )}{D _{0}} Q (s) d s, j = r j,

where

(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} η^{α β} h_{α β} .

Displayed formulas in this passage

IV-B@1#eq:NW · line 1120
$j (t) = cos (t D_{0}) A + \frac{sin ( t D _{0} )}{D _{0}} B + 2 \int_{0}^{t} \frac{sin (( t - s ) D _{0} )}{D _{0}} Q (s) d s, j = r j,$
unlabeled · line 1103
$(Δ^{- 1} b) (x) = - \frac{1}{4 π} \int \frac{b ( y )}{∣ x - y ∣} d y, ρ = Q_{00} (0), J_{i} = Q_{0 i} (0),$
unlabeled · line 1110
$a = - 2 Δ^{- 1} ρ, V_{i} = 2 Δ^{- 1} J_{i} .$
unlabeled · line 1114
$A_{00} = a, A_{0 i} = - V_{i}, A_{ij} = 0, B_{00} = - \partial_{i} V_{i}, B_{0 i} = B_{ij} = 0.$
unlabeled · line 1129
$(r h)_{μν} = h_{μν} - \frac{1}{2} η_{μν} η^{α β} h_{α β} .$

Nearby in this section

proposition: Proposition: Response and propagated constraint